This could depend on the circumstances you are locating. If you are talking with friends or in your family, and you all know the specific paper you are talking about, you can say " the paper", If you said "a paper", you mean one paper among several kinds of ,or many of papers, which you are uncertain. or you needn't speak out the paper's name.

1. I read a story in the newspaper. 2. I read a story in a newspaper. With the first, the speaker is referring to a specific newspaper. The listener knows by context which paper is being referred to. With the 2nd, the speaker isn't referring to a specific newspaper. Which newspaper it is isn't important to the speaker.

Both good. "the" is definite. The one newspaper that the listener knows about. "a" is indefinite, any old newspaper. . In practice there is little difference. If you are talking to people who read many newspapers, then "the newspaper" would be taken to mean a generic newspaper anyway and that you are not going to specify which paper, The Times, The Guardian, ...

in the newspaper ( anyone in general ) / in a newspaper ( a particular or specific one )